Lattices and $\ZZ$-modules in Euclidean space possess an infinitude
of subsets that are images of the original set under similarity
transformation. We classify such self-similar images according to
their indices for certain 4D examples that are related to 4D root
systems, both crystallographic and non-crystallographic. We
encapsulate their statistics in terms of Dirichlet series
generating functions and derive some of their asymptotic properties.